Zariski tangent space

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In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space, at a point P on an algebraic variety V (and more generally). It does not use differential calculus, being based directly on abstract algebra, and in the most concrete cases just the theory of a system of linear equations.

For example, suppose given a plane curve C defined by a polynomial equation

F(X,Y) = 0

and take P to be the origin (0,0). When F is considered only in terms of its first-degree terms, we get a 'linearised' equation reading

L(X,Y) = 0

in which all terms X^aY^b have been discarded if

a + b > 1.

We have two cases: L may be 0, or it may be the equation of a line. In the first case the (Zariski) tangent space to C at (0,0) is the whole plane, considered as a two-dimensional affine space. In the second case, the tangent space is that line, considered as affine space. (The question of the origin comes up, when we take P as a general point on C; it is better to say 'affine space' and then note that P is a natural origin, rather than insist directly that it is a vector space.)

It is easy to see that over the real field we can obtain L in terms of the first partial derivatives of F. When those both are 0 at P, we have a singular point (double point, cusp or something more complicated). The general definition is that singular points of C are the cases when the tangent space has dimension 2.

The definition given generalises directly to higher dimensions, in which case a number of equations may be involved in defining a variety V. The non-linear terms are dropped from all of them, giving a system of linear equations that define the tangent space. The definition of singular point is then that the dimension of the tangent space is greater than the dimension of V.

For more abstract theory, one notes that for any commutative local ring R, with maximal ideal M, there is the definition

M/M²

of an R-module, in terms of which the previous definitions can be recovered. For R coming from geometry over a field K, this will be a vector space over K. It therefore serves as an abstract analogue, and is also called the Zariski tangent space. It has an interpretation in terms of homomorphisms to the dual numbers for K,

K[t]/[t²]

which (thinking about affine schemes) allows one to speak in geometric terms, talking about tangent vectors.